Larry Riddle, Agnes Scott College

"It is this similarity between the whole and its parts, even infintesimal ones, that makes us consider this curve of von Koch as a line truly marvelous among all. If it were gifted with life, it would not be possible to destroy it without anihilating it whole, for it would be continually reborn from the depths of its triangles, just as life in the universe is."

E. Cesaro,

Atti d. R. Accademia d. Scienze d. Napoli,2, XII, number 15. [Excerpt from an article by Paul Lévy, reprinted in Edgar's textClassics on Fractals.]

Construction

Animation

Begin with a straight line (the blue segment in the top figure).
Divide it into three equal segments and replace the middle segment by
the two sides of an equilateral triangle of the same length as the
segment being removed (the two red segments in the middle figure).
Now repeat, taking each of the four resulting segments, dividing them
into three equal parts and replacing each of the middle segments by
two sides of an equilateral triangle (the red segments in the bottom
figure). Continue this construction.

The Koch curve is the limiting curve obtained by applying this construction an infinite number of times. For a proof that this construction does produce a "limit" that is an actual curve, i.e. the continuous image of the unit interval, see the text [4] by Edgar.

Function

System

IFS

Animation

The first iteration for the Koch curve consists of taking four
copies of the unit horizontal line segment, each scaled by **r** = 1/3.
Two segments must be rotated by 60°, one counterclockwise and one
clockwise. Along with the required translations, this yields the
following IFS.

\({f_1}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
1/3 & 0 \\
0 & 1/3 \\
\end{array}} \right]{\bf{x}}\) |
scale by r |

\({f_2}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
{1/6} & { - \sqrt 3 /6} \\
{\sqrt 3 /6} & {1/6} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}}
{1/3} \\
0 \\
\end{array}} \right]\) |
scale by r, rotate by 60° |

\({f_3}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
{1/6} & { \sqrt 3 /6} \\
{- \sqrt 3 /6} & {1/6} \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}}
{1/2} \\
\sqrt 3 / 6 \\
\end{array}} \right]\) |
scale by r, rotate by −60° |

\({f_4}({\bf{x}}) = \left[ {\begin{array}{*{20}{c}}
1/3 & 0 \\
0 & 1/3 \\
\end{array}} \right]{\bf{x}} + \left[ {\begin{array}{*{20}{c}}
{2/3} \\
0 \\
\end{array}} \right]\) |
scale by r |

The fixed attractor of this IFS is the Koch curve.

Angle 60

Axiom F

F —> F+F−−F+F

Axiom F

F —> F+F−−F+F

Dimension

The Koch curve is self-similar with 4 non-overlapping copies of itself, each scaled by the factor
**r** < 1. Therefore the similarity dimension, **d**, of the
attractor of the IFS is the solution to

\[\sum\limits_{k = 1}^4 {{r^d}} = 1 \quad \Rightarrow \quad d = \frac{{\log (1/4)}}{{\log (1/3 )}} = \frac{{\log (4)}}{{\log (3 )}} = 1.26186 \]

Properties

Koch constructed his curve in 1904 as an example of a
non-differentiable curve, that is, a continuous curve that does not have a
tangent at *any* of its points. Karl Weierstrass had first
demonstrated the existence of such a curve in 1872. The article by Sime Ungar provides a simple geometric proof.

The length of the intermediate curve at the *n*th iteration
of the construction is (4/3)^{n}, where *n* = 0 denotes the
original straight line segment. Therefore the length of the Koch
curve is infinite. Moreover, the length of the curve between any two
points on the curve is also infinite since there is a copy of the
Koch curve between any two points.

Three copies of the Koch curve placed outward around the three sides of an equilateral triangle form a simple closed curve that forms the boundary of the Koch snowflake. Three copes of the Koch curve placed so that they point inside the equilateral triangle create a simple closed curve that forms the boundary of the Koch anti-snowflake.

In the construction of the Koch curve, one can vary the size of the deleted section and one can also replace the equilateral triangle with a regular polygon with more sides. The construction of the (n,c)-Koch curve is described in the paper by Keleti and Paquette. Start with a closed line segment of length *L* and a positive number *c* less than 1. Replace the middle *cL* portion of the segment with the sides of a regular *n*-gon whose own sides are length *cL*.This results in *n*+1 new line segments. Apply the process recursively to each of these line segments, always gluing the regular *n*-gons outwards. The limit of this iterative process is the (n,c)-Koch curve. Three examples are given below. The paper by Keleti and Paquette investigates when the curve is self-intersecting or self-avoiding. A similar type of construction was described in the book by Gilbert Helmberg.

(4, 1/3)-Koch curve |
(5, 0.4)-Koch curve |

(6, 0.14)-Koch curve |

For a second variation, let K = {f_{1}, f_{2}, f_{3}, f_{4}} be the four functions in the Koch IFS. Let G be either the cyclic symmetry group Z_{n} or the dihedral symmetry group D_{n} for n ≥ 2. The group Z_{n} consists of counterclockwise rotations through angles that are multiples of 360°/n. The group D_{n} consists of symmetries of a regular polygon with n sides, including both the same rotations as in Z_{n} and also reflections. The one exception is D_{2} since there is no regular polygon with 2 sides. This group D_{2} is also known as the Klein four-group.

We can form a new IFS by applying each of the symmetries in G to each of the functions in K, i.e. form the set of affine transformations GK = { gf : g in G and f in K}. For example, if G = Z_{3}, then the three elements of G are the identity (no rotation), r_{120}, a counterclockwise rotation through 120°, and r_{240}, a counterclockwise rotation through 240°. The new IFS would then consist of the 12 functions {f_{k}, r_{120}f_{k}, r_{240}f_{k} for k = 1,2,3,4}. The first iteration would look like the following (using a horizontal line as the initial set as in the construction of the regular Koch curve).

This IFS again has a unique attractor, shown below, since each of the 12 affine transformations is contractive.

Notice that the self-similar parts of this fractal overlap, but that overall the fractal has three-fold rotational symmetry. If we rotate the fractal by 120°, then we are really just applying r_{120} to each of the 12 functions in the IFS. But this just permutes the elements of the IFS. For example, r_{120}r_{120}f_{k} is just the same as r_{240}f_{k} since two rotations through 120° is the same as one rotation through 240°. So if A is the attractor for the IFS, then
\[A = \bigcup\limits_{k = 1}^4 {\left( {{f_k}(A)\cup {{r_{120}}{f_k}(A)\cup {{r_{240}}{f_k}(A)} } } \right)} \]
and rotating A by 120° yields
\[\begin{align}
{r_{120}}(A) &= {r_{120}}\left( {\bigcup\limits_{k = 1}^4 {\left( {{f_k}(A)\cup {{r_{120}}{f_k}(A)\cup {{r_{240}}{f_k}(A)} } } \right)} } \right) \\
&= \bigcup\limits_{k = 1}^4 {\left( {{r_{120}}{f_k}(A)\cup {{r_{240}}{f_k}(A)\cup {{f_k}(A)} } } \right)} = A \\
\end{align}\]

The four elements of the group Z_{4} produces rotational symmetry with angles of 0°, 90°, 180°, and 270°. Applying these to the Koch curve iterated system produces the following fractal with four-fold rotational symmetry. It is colored using pixel counting (the color depends on how many times a particular pixel is plotted during the drawing of the fractal using the random chaos game algorithm).

Here is the fractal obtained using the dihedral group D_{4}. It is also colored using pixel counting. This fractal has both four-fold rotational symmetry (through rotations of 90°) as well as a reflective symmetry across horizontal, vertical, and diagonal lines. More details on symmetric fractals can be found here and in the book by Field and Golubitsky [15].

- Bannon, Thomas. "Fractals and Transformations," Mathematics Teacher, Vol. 84, No. 3 (March 1991), 178-185. [Accessible at JSTOR (subscription required)]
- Barcellos, Anthony. "The Fractal Geometry of Mandelbrot," The College Mathematics Journal, Vol. 15, No. 2, (1984), 98-114. [Available at the MAA website]
- Camp, Dane. "A Fractal Excursion," Mathematics Teacher, VOl. 84, No. 4 (April 1991), 265-275. [Accessible at JSTOR (subscription required)]
- Edgar, Gerald A.
*Measure, Topology, and Fractal Geometry,*Sringer-Verlag, 1990. - Edgar, Gerald A.
*Classics on Fractals,*Addison-Wesley 1993. Contains a translation of Koch's original article (see [8]). [Most of the chapter on Koch is viewable at Google Books] - Gardner, Martin. "Mathematical Games," Scientific American, April 1965, 128.
- Helmberg, Gilbert.
*Getting Acquainted with Fractals*, de Gruyter Publishing, 2007. [See Preview at Google Books] - Jones, Juw. "Fractals Before Mandelbrot-A Selective History,"
in
*Fractals and Chaos,*Crilly, Earnshaw, and Jones, Editors, Springer-Verlag 1991. - Koch, H. von. "Sur une courbe continue sans tangente, obtenue
par une construction géométrique élémentaire,"
*Arkiv for Matematik***1**(1904) 681-704. - Mandelbrot, Benoit.
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Fractals,",
*Byte,*August 1987, 123-132. - Peitgen, Heinz-Otto, Hartmut Jurgens and Dietmar Saupe.
*Fractals for the Classroom, Part One: Introduction to Fractals and Chaos,*Springer-Verlag New York, Inc. 1990. - Ungar, Sime. "The Koch Curve: A Geometric Proof," The American Mathematical Monthly, Vol. 114, No. 1 (January 2007), 61-66. [Available from JSTOR (subscription required)]
- Keleti, Tamás and Elliot Paquette. "The Trouble with von Koch Curves Built from
*n*-gons,", the American Mathematical Monthly, Vol. 117, No. 2 (February 2010), 124-137. An online supplement containing an interactive java applet is available at MathDL, the MAA Mathematical Sciences Digital Library. - Field, Michael and Martin Golubitsky.
*Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature*(2nd Edition), SIAM, 2009. [See Preview at Google Books. Chapter 7 is on Symmetric Fractals.]